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138                              THE  EFFECT  OF  JUNCTIONS  ON THE                            [3h
consider the present and a few analogous problems.     Some considerations a more general character are prefixed.
If P, Q, R be components of electromotive intensity, a, b, c those magnetisation, Maxwell's general circuital relations* for the dielectric give
da _ dQ    dR                                        /-, \
dt     dz     dy
and two similar equations, and
.-
dt         \dy    dz
also with two similar equations, V being the velocity of propagation.    Frc (1) and (2) we may derive
da    db     dc    A         dP    dQ    dR    A                       /<},
_  (-    1_      = o            |_ __.   _)_____ (j    ............... ^j
dec    dy    dz               dx     dy     dz
and, further, that        (^1_72V2VP, Q, R, a, b, c) = 0,   ..... ............... (4)
\Cit               /
where                      '         V2 = d*/dx2 + d^fdy* + d"jdz- ........................ (5)
At any point upon the surface of a conductor, regarded as perfect, t condition to be satisfied is that the vector (P, Q, R) be there normal. what follows we shall have to deal only with simple vibrations in which ; the quantities are proportional to ei:pt} so that d/dt may be replaced by ip.
It may be convenient to commence with some cases where the waves a in two dimensions (x, z) only, supposing that a, c, Q vanish, while b, P, are independent of y. From (1) and (2) we have
.
dx        dz
At the surface of a conductor P, Q are proportional to the directi cosines of the normal (n) ; so that the surface condition may be express simply by
l = . ..................................... 0
which, with           .            (-*+ + *.)_, .......................... .(,)
suffices to determine b.    In (7) k=p/V.    It will be seen that equations ( (7)  are  identical with  those  which  apply in  two  dimensions  to   aer vibrations  executed  in  spaces  bounded  by fixed walls, b  then  denoti velocity-potential.     When. 6 is  known, the remaining functions follow once.                               .            .            
*  Phil Trans. 1868 ;  Maxwell's Scientific Papers, Vol. n. p. 128.ub) tri               ,v .       .